How do you evaluate #\frac{3x ^ { 2} - 8x + 4}{x - 2}#?

1 Answer
Dec 8, 2017

#color(blue)(3x-2)#

Explanation:

We need to evaluate #color(red)((3x^2-8x+4)/(x-2))#

We will treat the given rational expression as follows:

Consider the Numerator first

#color(red)(3x^2-8x+4)# #.. color(red)(Expression.1)#

Evidently, this is a quadratic expression

We can factorize this quadratic expression:

To factor this quadratic expression, we will follow the procedure given below:

#color(green)(Step.1)#

We must split the coefficient of middle term into two numbers , such that when we add them we get the middle term, and when we multiply them we must get the product of the coefficient of the #x^2 term# and the constant,

Note that the product of the coefficient of the #x^2 term# and the constant is #(12)#,

#color(green)(Step.2)#

The two numbers are: #color(blue)(-6 and -2)#

When we add ( - 6) and ( -2 ) we get #(- 8)# and when we multiply the two values ( - 6) and ( -2 ) we get ( 3 X 4 = 12 )

Now, we write our #.. color(red)(Expression.1)# as follows:

#color(blue)(3x^2-2x - 6x+4)# #.. color(red)(Expression.2)#

#color(green)(Step.3)#

In this step, we break our #.. color(red)(Expression.2)# into groups:

#color(blue)(rArr (3x^2 - 2x) + (-6x +4))#

Factor out #color(green)(x)# from #color(blue)((3x^2 - 2x)# to obtain #color(blue)(rArr x*(3x - 2) )#

Factor out #color(green)(-2)# from #color(blue)((-6x +4)# to obtain #color(blue)(rArr -2*(3x -2) )#

#color(green)(Step.4)#

Using #color(green)(Step.3)# we can factor out the common term #color(blue)((3x-2)# and write the factors of our quadratic expression:

#color(blue)(rArr (3x -2) * (x - 2)#

#color(green)(Step.5)#

In this step we can rewrite our Numerator #color(red)(3x^2-8x+4)# as

#color(blue)(rArr (3x -2) * (x - 2)# #.. color(red)(Expression.3)#

#color(green)(Step.6)#

In this step, we will work on our given expression #color(red)((3x^2-8x+4)/(x-2))#

We can now write the above rational expression, using # color(red)(Expression.3)# as follows:

#color(blue)( {(3x -2) * (x - 2))/((x-2))}#

On simplification we get,

#color(blue)( {(3x -2) * cancel((x - 2)))/cancel((x-2))}#

Hence, #color(blue)(3x-2)# is our final answer.

I Hope this helps.